Geometry Lessons Research Articles

Effects of different ratios of worked solution steps and problem solving opportunities on cognitive load and learning outcomes

A crucial challenge for instructional designers is to determine the amount of support that is most beneficial for learning. This experiment investigated effects of different ratios of worked solution steps (high assistance) and to-be-solved problem steps (low assistance) on cognitive skill acquisition in geometry. High-school students ( N = 125) worked on a geometry lesson in a Cognitive Tutor under five different ratios (from zero worked steps and five to-be-solved steps to four worked steps and one to-be-solved step). Effects on cognitive load and learning outcomes were assessed. We expected the effectiveness of different ratios to vary with the type of learning outcomes (i.e., procedural vs. conceptual knowledge) and the difficulty of the to-be-learned principles. Results showed that for procedural knowledge (but not for conceptual knowledge) problem solving alone was most beneficial for the acquisition of procedural knowledge related to an easy principle. For a difficult principle, no ratio of worked steps and problem solving showed an advantage over another. Problem solving induced more extraneous load than studying worked examples. Thus, in determining optimal amounts of guidance type of knowledge and difficulty of the single to-be learned knowledge chunks should be considered.

Mathematical Exploration: Trapezoids to Triangles

Pattern blocks are often used in math class during geometry lessons to support concepts of shape, angle, tessellations, and symmetry. They are also used with fraction study (e.g., Lanius 2007; NCTM 2009a). Using pattern blocks to construct other larger geometric shapes can help students understand the relationships among various shapes. Algebraic relationships and functions emerge when students count blocks and measure figures. The activities in this article support the development of such geometric vocabulary as perimeter, area, sequence, and nonstandard measurement.

In the Flesh: Living Growing Conceptual Domains in a Geometry Lesson

A Response to Elizabeth Mowat & Brent Davis

The Emergence of 3D Geometry From Children's (Teacher-Guided) Classification Tasks

Geometry, classification, and the classification of geometrical objects are integral aspects of recent curriculum documents in mathematics education. Such curriculum documents, however, leave open how the work of classifying objects according to geometrical properties can be accomplished given that the knowledge of these properties is the planned outcome of the curriculum or lesson. The fundamental question of the present study therefore is this: How can a lesson in which children are asked to participate in a task of classifying regular 3-dimensional objects be a geometry lesson, given that the participating 2nd-grade children do not yet classify according to geometrical properties (predicates)? In our analyses, which are inspired by ethnomethodological studies of work, we focus on the embodied and collective work that leads to the emergence of the geometrical nature of this lesson. Thus, we report both the collective and the individual work by means of which the lesson outcomes—the complete classification of a set of “mystery” objects according to geometrical (shape) rather than other (color, size, “pointy-ness”) properties—are achieved. In the process, our study shows how geometrical work is reproduced by 2nd-grade children who, in a division of labor with their teachers, produce a particular set of geometrical practices (sorting three-dimensional objects according to their geometrical properties) for the 1st time.

The Prankquean and the Localization of Legend

The Prankquean and the Localization of Legend Vincent Deane (bio) Time and the river and the mountain are the real heroes of my book. ... Yet the elements are exactly what any novelist might use: man and woman, birth, childhood, night, sleep, marriage, prayer, death. ... There is nothing paradoxical about this. ... Only I am trying to build many planes of narrative with a single esthetic purpose. ... Did you ever read Laurence Sterne ...?1 The story of the prankquean and Jarl van Hoother is one of the most widely analyzed passages in Finnegans Wake and it is easy to see why. Coming near the beginning of the book (FW 21.05-29.36), it is a manageably short and self-contained set piece. Even if we cannot gloss many of the words we are carried along by what has all the appearances of a racy anecdote full of tart exchanges between its leading characters. Unlike the 'bluddle filth' (FW 10.08-9) of the 'Museyroom' episode that was written at the same time (FW 8.09-10.23), the mysteries of the prankquean story's content are offset by a reassuringly graspable form and a narrative that sticks fairly closely to its original plot, which is more than can be said for the supine meanderings of the mistress Kathe. Yet it seems to me that some of the most important aspects of the story and its characters have been insufficiently dealt with by previous commentators. At the local level, there are the names of those involved. We have a quite satisfactory explanation for the jiminies (Latin gemini, twins),2 a partial one for Tristopher and Hilary (why 'Tristopher'?),3 but what about the names given to the prankquean and van Hoother? Why does the prankquean urinate each time she visits Jarl van Hoother's keep? Or does she? And why does Jarl van Hoother defecate at the end of the story? Or does he? Readers might well imagine that such matters, along with the prankquean's riddle, which has been dyed with countless hues and slipped under all manner of microscopes, would have been fairly well exhausted by now, but some of the most simple [End Page 30] and basic elements have not been explained. At a more general level I hope to throw some new light on the function of the story within Book I, Chapter 1 and against the larger framework of Finnegans Wake as a whole. Readers of the Wake are often rightly suspicious of the use of Joyce's manuscripts and notebooks as tools for exegesis. Too often this leads to a reduction of the richly ambiguous and allusive text of the book to a Skeleton Key-like narrative, or to the application of otherwise irretrievable glosses to words and passages within the book. I hope to avoid these pitfalls. Through a detailed account of the way that Joyce built up the episode, my intention is to make more visible the different metaphorical layers that he has superimposed in the finished text; if I privilege some of these layers it is not to deny the role of the others. Most of the notebook-based material will be self-evident. Work in Progress 1923-6 By the time he started writing the 'prankquean', Joyce had written all the chapters of Book I except the first and sixth; the 'Four Watches of Shaun', which make up Book III; as well as 'The Triangle' (also known as 'the geometry lesson'), which forms part of Book II, Chapter 2 (FW 282.05-304.04). In the process he had gradually arrived at the five members of the Earwicker family in Chapelizod through a complicated and extremely varied series of sketches derived from myth, history, and popular culture. The Mark-Tristan-Isolde triangle was the starting point for HCE, Issy, and the idea of a rebellious, treacherous son figure, as well as the location of the family home in Chapelizod. The Mark aspect of HCE, which had already been prefigured in Joyce's very first sketch for his new book — a portrait of Roderick O'Conor, 'the last preelectric King of all Ireland' (see FW 380.12-13) — was further developed by means...

Geometrical analogies in mathematics lessons

A typical form of thinking to approach problem solutions humanly is thinking in analogous structures. Therefore school, especially mathematical lessons should help to form and to develop corresponding heuristic abilities of the pupils. In the contribution, a summary of possibilities of mathematics lessons regarding this shall particularly be conveyed from the view of geometry lessons in different age groups, being supposed to be exemplarily illustrated introducing character and object of making analogies as an example. Different kinds of analogous conclusions are described exemplarily with some examples for different geometrical contents and didactic situations.

By Way of Introduction

“Measurement, the Forgotten Strand” was the title of a presentation given by Glenda Lappan, past president of the National Council of Teachers of Mathematics, at the 79th Annual Meeting in Orlando in April 2001. What an apt title! Measurement is a concept that, if not completely forgotten, is often taken for granted. Yet measurement is an integral part of all mathematics content as well as the content of most other subjects. Measurement is central in geometry lessons as well as in science. Despite their wide application, however, measurement lessons are often found at the end of traditional textbooks—in chapters that most teachers do not have time to teach or are insufficiently addressed. This fact is troubling when we stop to consider how completely measurement is integrated into our lives.

How much time do students have to think about teacher questions? An investigation of the quick succession of teacher questions and student responses in the German mathematics classroom

Several studies have shown that the style of the German mathematics classroom at secondary level is mostly based on the so called “fragend-entwickelnde” teaching style which means developing the lesson content by a teacher directed sequence of teacher questions and student responses. In this article we describe a study on the time the students have for thinking about a teacher question in the public classroom interaction. Our investigation is based on a reanalyasis of 22 geometry lessons from grade 8 classes which mainly deal with a challenging proving content. The results show that the average time between a teacher question and a student response is 2.5 seconds. There are no remarkable differences between different phases of the lessons like comparing homework, repetition of content or working on new content. Moreover, for 75% of the teacher questions the first student was called to answer within a three second time interval.

Developing geometry thinking through multimedia learning activities

In this study, a multimedia learning software program named GeoCAL is described. It is based on van Hiele's geometric thinking level theory, which consists of four levels: recognition, visual association, description/analysis, and abstraction/relation. In addition to presenting the software design, this study also explores the learning effects of GeoCAL on each of the geometric thinking levels and on overall geometric thinking. The subjects of this study are second-grade elementary school students of an average age of eight who have not previously had formal lessons in geometry. The experimental results indicate that, with the exception of recognition ability, GeoCAL produces significant learning effects on visual association, description/analysis and abstraction/relation as well as overall geometric thinking.

The computer as an aid in the development of geometrical proficiency: a differential approach

The efficiency and modes of usage of computerized tools (DGE and some other types of tools) were studied in relation to: students’ abilities; the routineness of a problem; and the type of course (all three parameters defined in the paper). It was found that there was no significant difference in modes of computer usage for routine problems; a significant difference appeared only in the group where the students had been taught to use these tools ‘in real time’, i.e. when it was found appropriate by the lecturer during a geometry lesson. Besides that it was found that the percentage of high-level students who improved their results with non-routine problems due to use of the computer is significantly higher than that of lower-level students.

Introducing Russell Graphically

Reviews 75 INTRODUCING RUSSELL GRAPHICALLY Stefan Andersson Theology and Religious Studies / U. of Lund s–223 62, Lund, Sweden butterflyandeson@hotmail.com Dave Robinson and Judy Groves. Introducing Bertrand Russell. Cambridge, uk: Icon Books Ltd., 2002. Pp. 176. us$11.95; cdn$16.99; £9.99. ntil now there have appeared at least 20 short books in different languages Uwith the aim of introducing the life and work—or both—of Bertrand Russell . With the publication of Introducing Bertrand Russell written by Dave Robinson , illustrated by Judy Groves and edited by Richard Appignanesi we have the first attempt to introduce the life and thought of Russell with the help of 76 Reviews cartoon-like illustrations and easy-to-follow diagrams. This particular genre of presenting a topic or a thinker started some 30 years ago, but it is not until now that Russell has been blessed with an attempt to present the most important aspects of his life and thinking in this particular way. After first telling us a little bit about Russell’s upbringing and his fear of madness, Robinson in Introducing Bertrand Russell uses Russell’s passionate search for certainty in mathematics as the starting-point for his description of Russell’s philosophical odyssey. The 170 pages are divided into 117 short sections in which the author tries to convey the most interesting aspects of Russell’s life and thinking. Starting with the fourth “The Geometry Lesson” and ending 26 sections later with “What is Logic?”, where Russell reluctantly admits that “Logic, in other words, is an empty process of linguistic analysis” (p. 43), we get a quick but fair description of the development of Russell’s thinking about mathematics and logic. The rest of the book is much more impressionistic and jumps from one topic to another with Wittgenstein showing up in different parts of the book without being allowed to steal the show. Robinson’s intentions have obviously been to give a brief introduction to and assessment of Russell’s thinking and political activities, but the book starts with a well-intended exaggeration: “Everyone has heard of Bertrand Russell. He was a great thinker, an agitator imprisoned for his beliefs, and a man who changed Western philosophy for ever” (p. 3). Unfortunately, far from everyone has heard about Bertrand Russell, but what follows is true. What surprises me is that a person who holds Russell in such high esteem should not be able to get some fundamental facts right. The work on publishing The Collected Papers of Bertrand Russell is being pursued at McMaster University in Hamilton and not at some non-existing institution called MacMaster University in Toronto. Russell’s famous essay “A Free Man’s Worship” was not written and published in 1923 for the first time, but twenty years earlier. Robinson describes it as a “kind of gospel to agnosticism” (p. 143), which is not totally misleading, but by misdating it by twenty years, he creates a very confusing picture of the development of Russell’s personal religious beliefs as well as his views on religion. The section “Russell on Religion” (pp. 142–3) starts with the sentence: “Russell lost his Christian faith at an early age but often admitted to mysterious spiritual longings.” This is vague, but it is acceptable. Then we get to know a little about “a personal epiphany” that Russell had on 10 February 1901. Russell used different expressions to refer to this very important experience of “mystic illumination” or even “first conversion”. The ballooncontains—asoften—adirectquotationfromRussell;inthiscase,thefirst volume of his Autobiography, where he talks about the insights that he gained Reviews 77 from this experience: “that the loneliness of the human soul is unendurable; nothing can penetrate it except the highest intensity of the sort of love that religious teachers have preached.” This description of Russell’s most important “religious”, “mystical” or “spiritual” experience was actually written in 1931. Robinson seems happily unaware of all these things. The confusion gets worse when we turn to the following section, “No Proof or Disproof of God”, where the first sentence states: “Russell had not lost his faith in Christianity but had long stopped believing in God...

From Description to Proof

These quotes came from sixth-grade students during several geometry lessons. Because of my interest in how students stimulate one another's reasoning in mathematics, I spent some time in classrooms with teachers who were willing to step back and let their students express their thinking, and whose careful modeling would resonate in the students' reasoning at a later time. The two vignettes that follow illustrate the fact that student-to-student discourse and careful teacher modeling support a transition path to more formal mathematical reasoning.

Research, Reflection, Practice: Implications of Research on Children's Understanding of Geometry

In my mathematics methods course for preservice teachers, I ask my students to assess a group of elementary school students concerning their level of geometric reasoning. To do so, they use tasks that focus on assessing and extending students' geometric understandings. These open-ended tasks, along with a framework developed from research findings involving children's geometric reasoning, are described in this article. An important aspect of these tasks is that they focus on how students communicate their reasoning so that my preservice teachers can make more informed instructional decisions when planning a follow-up geometry lesson. Research on geometric reasoning has shown that a match between students' reasoning level and instructional tasks is crucial if meaningful learning is to occur (Crowley 1987).

The Use of Dot Paper in Geometry Lessons

One of the most signjficant documents written in the last twenty years is the NCTM publication, Curriculum and Evaluation Standards for School Mathematics (1989).

On squares, rectangles, rhombuses, ... and the influence of culture and language on students’ conceptions

In this paper we discuss the influence of cultural and linguistic factors on students’ understanding of some special types of quadrilaterals which are common in geometry lessons, namely the square, the rectangle, the rhombus, the parallelogram, and the trapezium. In this direction we examine a number of clinical interviews with 10th grade students. In particular we analyse, from a culture and linguistic viewpoint, the students’ difficulties with respect to the logical/conceptual structure of the above types of quadrilaterals.

Build a City

Could I find a new angle for the old geometry lesson? This thought cro sed my mind as I prepared to teach a geometry unit to my class of fifth and ixth graders. The more I pored through old copies of teaching magazines and books, the more I became convinced that to give my students some tangible evidence that geometry is alive today in various careers. I would have to create the experience I ought for them. As a result, J designed a week-long build-acity project to let my students become familiar with the history of the five Platonic solids and then use these solid to create a city.

Plato, Leibniz, and the Furnished Soul

Plato, Leibniz, and the Furnished Soul' GRAEME HUNTER and BRAD INWOOD IT IS A TRADITIONAL and perennially important question whether there are truths which the soul already has when a human being is born. Plato and Leibniz have given two of the most influential affirmative answers. ~ Characteristically , Leibniz's version is not a rote assimilation of the Platonic doctrine , but a critical reaction to it. A comparative examination of their approaches to the question, whether man is born with knowledge in the soul, recommends itself. We look first at Plato's theory of recollection, its motivations and the arguments for it. Leibniz's doctrine of innate ideas is then considered with particular attention to his disagreements with Plato. The discussion isolates enduring features of this type of theory as well as shedding light on the contrasting philosophical methods of Plato and Leibniz. PART 1 Plato's best-known discussion of learning as recollection is in the Meno (8od-86c), where he argues against a famous paradox which purports to Plato will be cited according to Stephanus pages, from the Oxford ClassicalText, Leibniz according to the following key: A: G. W. Leibniz: SaemtlicheSchriften und Briefe. Berlin: AkademieVerlag . Series began publishing 1923. C: G. W. Leibniz: Opuscules et fragments in~dits, ed. L. Couturat. Hildesheim: Olms, 1966. Reprint of Paris, ]9o3 . DM: Discours de Metaphysique in GP IV, 427-63. F de C: G. W. Leibniz: Nouvelles lettreset opuscules in~ditsdeLeibniz, ed. A. Foucher de Careil. Paris: Durand, 1857. GM: G. W. Leibniz: Die mathematischen Schriften, 7 vols. ed. C. I. Gerhardt. Hildesheim: Olms, 1971. Reprint of Halle, 1848-65. GP: G. W. Leibniz: Diephilosophischen Schriften, 7 vols., ed. C. I. Gerhardt. Hildesheim: Olms, 196o-61. Reprint of Berlin, 1875-189o . NE: Nouveaux essaissur l'entendement humain in A VI, vi. 2 A more complete survey of the doctrine of innate ideas would of course have to deal with Descartes in particular. However, it is clear that Leibniz derived his inspiration primarily from Plato, not Descartes, whose doctrine of innate ideas he regarded as ill-conceivedand misleading (NEII, xxi, 73; A. Robinet, "Grundprobleme der 'Nouveaux Essais' " in Studia Leibnitiana Supplementa vol. 3 (1969) 2o-33, esp 95). [423] 424 JOURNAL OF THE HISTORY OF PHILOSOPHY 22:4 OCT i984 show that one cannot learn anything. "Meno's Paradox," as it is now known, states that one cannot learn what one does not already know (for one would not recognize it even if one found it) and that one cannot learn what one does know (for that would not count as learning). Socrates begins his argument with a reference to the traditional religious doctrine that the soul is immortal, has been born many times and has learned everything there is to learn during its sojourns in the underworld and during its previous corporeal existences. The philosophical core of his position, however, is revealed in his examination of the slave-boy. Socrates elicits a geometrical theorem from an untutored slave, claiming that it was drawn from the slave's own intellectual resources and was not taught to him, but only "stirred up" by Socratic questioning. The geometry lesson itself is well-known. At the end of it Meno admits that the slave-boy had in his soul true opinions about geometry all along and that his soul acquired these before he was born (85b-86b). Socrates himself does not endorse these conclusions unequivocally, for he concludes this part of the discussion with a remarkable disclaimer. "There are other points about the argument which I would not assert with complete confidence. But that we would be better men and more courageous and less slothful if we believed that one must seek for what one does not know than if we believed that it is neither possible to discover nor necessary to seek for what we do not understand, this I would fight for, if I were able, both in word and in deed." Socrates's disclaimer correctly suggests that one should consider carefully how strong a conclusion can be drawn from the slave-boy experiment and the ensuing discussion with Meno. The strongest claim would be that (1) All knowledge is acquired before this birth...

A Geometry lesson from National Assessment

The geometry exercises administered during the 1977-78 mathematics assessment of the National Assessment of Educational Progress (N AEP) focused primarily on concepts of informal geometry rather than the more formal deductive geometry typically taught in a high school course. Results did show that students who studied formal geometry performed better.

The Philosophical Sense of Theaetetus' Mathematics

Previous articleNext article No AccessThe Philosophical Sense of Theaetetus' MathematicsM. F. BurnyeatM. F. Burnyeat Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Isis Volume 69, Number 4Dec., 1978 Publication of the History of Science Society Article DOIhttps://doi.org/10.1086/352111 Views: 24Total views on this site Citations: 16Citations are reported from Crossref Copyright 1978 History of Science Society, Inc.PDF download Crossref reports the following articles citing this article:Vincenzo Vullo The First Scientific Age and Birth of the Science: From the Beginning of Hellenism to the Diaspora of Scientists of the Museum of Alexandria, (Feb 2020): 21–56.https://doi.org/10.1007/978-3-030-40164-1_3VICTOR PAMBUCCIAN THE ARITHMETIC OF THE EVEN AND THE ODD, The Review of Symbolic Logic 9, no.22 (Jan 2016): 359–369.https://doi.org/10.1017/S1755020315000386J. Lennart Berggren History of Greek Mathematics: A Survey of Recent Research [1984], (Sep 2013): 3–15.https://doi.org/10.1007/978-3-642-36736-6_1Tamer Nawar Knowledge and True Belief at Theaetetus 201a–c, British Journal for the History of Philosophy 21, no.66 (Dec 2013): 1052–1070.https://doi.org/10.1080/09608788.2013.822344S. Pollard Dianoia Left and Right, Philosophia Mathematica 21, no.33 (Apr 2013): 309–322.https://doi.org/10.1093/philmat/nkt010H. H. Benson The Problem is not Mathematics, but Mathematicians: Plato and the Mathematicians Again, Philosophia Mathematica 20, no.22 (Jan 2011): 170–199.https://doi.org/10.1093/philmat/nkq029Luigi Borzacchini Incommensurability, Music and Continuum: A Cognitive Approach, Archive for History of Exact Sciences 61, no.33 (Mar 2007): 273–302.https://doi.org/10.1007/s00407-007-0125-0András Máté Árpád Szabó and Imre Lakatos, or the Relation Between History and Philosophy of Mathematics, Perspectives on Science 14, no.33 (Sep 2006): 282–301.https://doi.org/10.1162/posc.2006.14.3.282F. Acerbi A Reference to Perfect Numbers in Plato?s Theaetetus, Archive for History of Exact Sciences 59, no.44 (Feb 2005): 319–348.https://doi.org/10.1007/s00407-004-0094-5Benno Artmann A proof for Theodorus' theorem by drawing diagrams, Journal of Geometry 49, no.1-21-2 (Mar 1994): 3–35.https://doi.org/10.1007/BF01228045Jens Høyrup Dýnamis, the Babylonians, and Theaetetus 147c7–148d7, Historia Mathematica 17, no.33 (Aug 1990): 201–222.https://doi.org/10.1016/0315-0860(90)90001-TR. A. H. Waterfield Emendations of [Iamblichus], Theologoumena Arithmeticae (De Falco), The Classical Quarterly 38, no.11 (Feb 2009): 215–227.https://doi.org/10.1017/S0009838800031426J.L Berggren History of Greek mathematics: A survey of recent research, Historia Mathematica 11, no.44 (Nov 1984): 394–410.https://doi.org/10.1016/0315-0860(84)90024-7Alan C Bowen The beginnings of Greek mathematics, Historia Mathematica 11, no.33 (Aug 1984): 335–345.https://doi.org/10.1016/0315-0860(84)90045-4A. G. MOLLAND Á. Szabó, The Beginnings of Greek Mathematics, The British Journal for the Philosophy of Science 32, no.33 (Dec 2020): 306–309.https://doi.org/10.1093/bjps/32.3.306Gustav Adolf Seeck Literaturhinweis, (): 161–164.https://doi.org/10.4000/books.chbeck.1362

Patterns of intersection

An interesting pattern of intersections was discovered by an eighth-grade student after an introduction to a geometry lesson on how lines in a plane may be associated.